Differentiable Manifolds#

Definition: Smooth Compatibility

Let \((\mathcal{D}_{\varphi}, \varphi)\) and \((\mathcal{D}_{\psi}, \psi)\) be charts on a manifold.

We say that \((\mathcal{D}_{\varphi}, \varphi)\) and \((\mathcal{D}_{\psi}, \psi)\) are smoothly compatible if the transition map \(\psi \circ \varphi^{-1}\) is a diffeomorphism or \(\mathcal{D}_{\varphi} \cap \mathcal{D}_{\psi} = \varnothing\).

Definition: Smooth Atlas

An atlas for a manifold is smooth if all of its charts are pairwise smoothly compatible.

Definition: Differentiable Manifold

A differentiable manifold or smooth manifold \((M, \mathcal{A})\) is a manifold \(M\) equipped with a maximal smooth atlas \(\mathcal{A}\) on it.

We call \(\mathcal{A}\) the differentiable structure or smooth structure of \((M, \mathcal{A})\).